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Theorem atpointN 33387
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a  |-  A  =  ( Atoms `  K )
ispoint.p  |-  P  =  ( Points `  K )
Assertion
Ref Expression
atpointN  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )

Proof of Theorem atpointN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . 4  |-  { X }  =  { X }
2 sneq 3887 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
32eqeq2d 2454 . . . . 5  |-  ( x  =  X  ->  ( { X }  =  {
x }  <->  { X }  =  { X } ) )
43rspcev 3073 . . . 4  |-  ( ( X  e.  A  /\  { X }  =  { X } )  ->  E. x  e.  A  { X }  =  { x } )
51, 4mpan2 671 . . 3  |-  ( X  e.  A  ->  E. x  e.  A  { X }  =  { x } )
65adantl 466 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  E. x  e.  A  { X }  =  {
x } )
7 ispoint.a . . . 4  |-  A  =  ( Atoms `  K )
8 ispoint.p . . . 4  |-  P  =  ( Points `  K )
97, 8ispointN 33386 . . 3  |-  ( K  e.  D  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
109adantr 465 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
116, 10mpbird 232 1  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   {csn 3877   ` cfv 5418   Atomscatm 32908   PointscpointsN 33139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-pointsN 33146
This theorem is referenced by: (None)
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